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Galois Representations and Modular Forms

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Galois Representations and Modular Forms Galois representations and modular forms Takeshi Saito July 17-22, 2006 at IHES Abstract This note is based on a series of lectures given at the summer school held on July 17-29, 2006 at IHES. The purpose of the lectures is to explain the basic ideas in the geometric construction of the Galois representations associated to elliptic modular forms of weight at least 2. Motivation The Galois representations associated to modular forms play a central role in the modern number theory. In this introduction, we give a reason why they take such a position. A goal in number theory is to understand the finite extensions of Q. By Galois Q Q theory, it is equivalent to understand the absolute Galois group GÉ = Gal( / ). One may say that one knows a group if one knows its representations. Representations are classified by the degrees. The class field theory provides us a precise understanding of the representations of degree 1, or characters. By the theorem → C× of Kronecker-Weber, a continuous character GÉ is a Dirichlet character → Q Q → Z Z × → C× GÉ Gal( (ζN )/ ) ( /N ) for some integer N ≥ 1. If we consider not only complex continuous characters but → Q× also -adic characters GÉ for a prime , we find more characters. For example, the -adic cyclotomic character is defined as the composition: n × × × → Q n ∈ N Q Q n Q → Z Z Z ⊂ Q GÉ Gal( (ζ ,n )/ ) =← lim− nGal( (ζ )/ ) ←lim− n( / ) = . The -adic characters “with motivic origin” are generated by Dirichlet characters and -adic cyclotomic characters: { } “geometric” -adic character of GÉ = Dirichlet characters, -adic cyclotomic characters if we use a fancy terminology “geometric”, that will not be explained in this note. For the definition, we refer to [13]. 1 When we leave the realm of class field theory, the first representations we encounter are those of degree 2. For -adic Galois representation of degree 2, we expect to have (cf. [13]) a similar equality { } odd “geometric” -adic representation of GÉ of degree 2 of distinct Hodge-Tate weight = {-adic representation associated to modular form of weight at least 2}, up to twist by a power of the cyclotomic character. In other words, the Galois repre- sentations associated to modular forms are the first ones we encounter when we explore outside the domain of class field theory. In this note, we discuss only one direction ⊃ established by Shimura and Deligne ([21], [7]). We will not discuss the other direction ⊂, which is almost established after the revolutionary work of Wiles, although it has significant consequences including Fermat’s last theorem, the modularity of elliptic curves, etc. ([24], [1]). In Section 1, we recall the definition of modular forms and state the existence of Galois representations associated to normalized eigen cusp forms. We introduce C Z 1 modular curves defined over and over [ N ] as the key ingredient in the construction of the Galois representations, in Section 2. Then, we construct the Galois representations in the case of weight 2 by decomposing the Tate module of the Jacobian of a modular curve in Section 3. In the final Section 4, we briefly sketch an outline of the construction in the higher weight case. Proofs will be only sketched or omitted mostly. The author apologizes that he also omits the historical accounts completely. The author would like to thank the participants of the summer school for pointing out numerous mistakes and inaccuracies during the lectures. Contents 1 Galois representations and modular forms 3 1.1 Modular forms . .............................. 3 1.2 Examples .................................. 4 1.3 Hecke operators . .............................. 6 1.4 Galois representations ........................... 7 2 Modular curves and modular forms 8 2.1 Elliptic curves . .............................. 8 2.2 Elliptic curves over C ............................ 10 2.3 Modular curves over C ........................... 11 2.4 Modular curves and modular forms . ................... 14 Z 1 2.5 Modular curves over [ N ] ......................... 15 2.6 Hecke operators . .............................. 17 2 3 Construction of Galois representations: the case k =2 19 3.1 Galois representations and finite ´etale group schemes . ........ 19 3.2 Jacobian of a curve and its Tate module ................. 19 3.3 Construction of Galois representations .................. 22 3.4 Congruence relation ............................. 23 4 Construction of Galois representations: the case k>2 25 4.1 Etale cohomology .............................. 25 4.2 Construction of Galois representations .................. 27 References 27 1 Galois representations and modular forms 1.1 Modular forms Let N ≥ 1 and k ≥ 2 be integers and ε :(Z/N Z)× → C× be a character. We will define C-vector spaces Sk(N,ε) ⊂ Mk(N,ε) of cusp forms and of modular forms of level N, weight k and of character ε. We will see later in §3.4 that they are of finite dimension by using the compactification of a modular curve. For ε = 1, we write Sk(N) ⊂ Mk(N) for Sk(N,1) ⊂ Mk(N,1). For a reference on this subsection, we refer to [12] Chapter 1. A subgroup Γ ⊂ SL2(Z) is called a congruence subgroup if there exists an integer N ≥ 1 such that Γ ⊃ Γ(N) = Ker(SL2(Z) → SL2(Z/N Z)). In this note, we mainly consider the congruence subgroups ab Γ (N)= ∈ SL (Z) a ≡ 1,c≡ 0modN 1 cd 2 ab ⊂ Γ (N)= ∈ SL (Z) c ≡ 0modN . 0 cd 2 ab We identify the quotient Γ (N)/Γ (N) with (Z/N Z)× by → d mod N. The 0 1 cd indices are given by 1 [SL (Z):Γ(N)] = (p +1)pordp(N)−1 = N 1+ , 2 0 p p|N p|N 1 [SL (Z):Γ(N)] = (p2 − 1)p2(ordp(N)−1) = N 2 1 − . 2 1 p2 p|N p|N The action of SL2(Z) on the Poincar´e upper half plane H = {τ ∈ C|Im τ>0} is defined by aτ + b γ(τ)= cτ + d 3 ab for γ = ∈ SL (Z) and τ ∈ H. For a holomorphic function f on H, we define cd 2 ∗ a holomorphic function γkf on H by 1 γ∗f(τ)= f(γτ). k (cτ + d)k ∗ ∗ If k = 2, we have γ (fdτ)=γ2 (f)dτ. Definition 1.1 Let Γ ⊃ Γ(N) be a congruence subgroup and k ≥ 2 be an integer. We say that a holomorphic function f : H → C is a modular form (resp. a cusp form) of weight k with respect to Γ, if the following conditions (1) and (2) are satisfied. ∗ ∈ (1) γkf = f for all γ Γ. ∗ ∗ ∗ (2) For each γ ∈ SL2(Z), γ f satisfies γ f(τ + N)=γ f(τ) and hence we have k ∞ k k ∗ n ∗ n τ a Fourier expansion γkf(τ)= n=−∞ a N (γkf)qN where qN = exp(2πiN ). We require the condition ∗ n a N (γkf)=0 be satisfied for n<0 (resp. n ≤ 0) for every γ ∈ SL2(Z). We put { | } Sk(Γ) = f f is a cusp form of weight k w.r.t. Γ ⊂ { | } Mk(Γ) = f f is a modular form of weight k w.r.t. Γ and define Sk(N)=Sk(Γ0(N)). Since Γ0(N) contains Γ1(N) as a normal subgroup, the → ∗ group Γ0(N) has a natural action on Sk(Γ1(N)) by f γkf. Since Γ1(N) acts trivially × on Sk(Γ1(N)), we have an induced action of the quotient Γ0(N)/Γ1(N)=(Z/N Z) on × Sk(Γ1(N)). The action of d ∈ (Z/N Z) on Sk(Γ1(N)) is denoted by d and is called the diamond operator. The space Sk(Γ1(N)) is decomposed by the characters Sk(Γ1(N)) = Sk(N,ε) × × → ε:(/N ) × where Sk(N,ε)={f ∈ Sk(Γ1(N))|df = ε(d)f for all d ∈ (Z/N Z) }. The fixed part Γ0(N) Sk(Γ1(N)) = Sk(N,1) is equal to Sk(N)=Sk(Γ0(N)). 1.2 Examples We give some basic examples following [19] Chapter VII. First, we define the Eisenstein series. For an even integer k ≥ 4, we put 1 Gk(τ)= k . (mτ + n) m,n∈,(m,n)=(0,0) It is a modular form of weight k. 4 The q-expansion of an Eisenstein series is computed as follows. The logarithmic ∞ τ 2 derivative of sin πτ = πτ 1 − gives n2 n=1 ∞ ∞ 1 1 1 1 −2πi + qn = + + . 2 τ τ + n τ − n n=1 n=1 d 1 d Applying k − 1-times the operator q = , one gets dq 2πi dτ ∞ (−1)k(k − 1)! 1 nk−1qn = . (2πi)k (τ + n)k n=1 n∈ ≥ k−1 For k 4 even, by putting σk−1(n)= d|n d and ∞ 2 n E (q)=1+ σ − (n)q ∈ Q[[q]], k ζ(1 − k) k 1 n=1 we deduce (k − 1)! (k − 1)! G (τ)= (2ζ(k)+(G (τ) − 2ζ(k))) (2πi)k k (2πi)k k ∞ n = ζ(1 − k)+2 σk−1(n)q = ζ(1 − k)Ek(q). n=1 Recall that the special values of the Riemann zeta function at negative odd integers 1 1 1 ζ(−1) = − ,ζ(−3) = ,ζ(−5) = − , ... 12 120 252 are non-zero rational numbers (cf. [19] p.71, Chapter VII Proposition 7). The C-algebra ∞ of modular forms of level 1 are generated by the Eisenstein series: k=0 Mk(1) = C[E4,E6] (cf. loc. cit. Section 3.2). The Delta-function defined by ∞ ∞ 1 Δ(q)= (E3 − E2)=q (1 − qn)24 = τ(n)qn 123 4 6 n=1 n=1 is a cusp form of weight 12 and of level 1 (see [19] Chapter VII Sections 4.4, 4.5).
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